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Conversion Abilities are abilities that will give you full control of a hostile unit. If they are still under your control at the end of the battle, they will turn to your side permanently.
The only kind of Conversion that cannot have permanent effect is the spell Arcane Binding

Damage abilities are those used to kill hostile units. First, there are melee attacks, like Melee Strike, and ranged abilities. Ranged abilities occur in three main classes: Projectile based (Shoot Bow), which generally suffer a range and/or obstacle penalty, abilities that occur on a location, like Call Lightning, which have no trajectory. Lastly, there are the Breath Type attacks, which will suffer obstacle penalties, but not range, and which can interact in a unique way in some cases, like them not being affected by Blind, all of them able to be used when engaged, and hitting all units in the general direction, generally a sixth of the area that could be possibly affected. These occur only on short or medium range.

Debuffs are abilities which lower the enemy's statistics, or otherwise hinder their combat effectiveness. There are the two main types of disables(Stunning Touch), or just lowering of statistics(Immolating Touch)